Subgroups — C₅×C₁₃×Dic₁₂₃ #44544285

Order 31980.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₁₃×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₅×C₁₃×Dic₄₁	—	3	1	no	maximal, Hall
3	C₃×C₅×C₁₃×C₄₁	C₄	1	1	yes	characteristic, complemented, Hall
4	C₁₃×Dic₁₂₃	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂×C₅×C₁₃×C₄₁	D₃	1	1	yes	characteristic
6	C₂×C₃×C₁₃×C₄₁	C₂×C₅	1	1	yes	characteristic
7	C₅×C₁₃×C₄₁	Dic₃	1	1	yes	characteristic, complemented, Hall
8	C₅×Dic₁₂₃	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₁₃×Dic₄₁	—	3	1	no	Hall
10	C₃×C₁₃×C₄₁	C₄×C₅	1	1	yes	characteristic, complemented, Hall
11	C₂×C₃×C₅×C₄₁	C₂×C₁₃	1	1	yes	characteristic
12	C₂×C₁₃×C₄₁	C₅×D₃	1	1	yes	characteristic
13	C₅×Dic₄₁	—	3	1	no	Hall
14	C₅×Dic₃×C₁₃	—	41	1	no	maximal, Hall
15	C₃×C₅×C₄₁	C₄×C₁₃	1	1	yes	characteristic, complemented, Hall
16	C₁₃×C₄₁	C₅×Dic₃	1	1	yes	characteristic, complemented, Hall
17	Dic₁₂₃	C₅×C₁₃	1	1	yes	characteristic, direct product factor, Hall
18	C₂×C₅×C₄₁	D₃×C₁₃	1	1	yes	characteristic
19	C₂×C₃×C₅×C₁₃	D₄₁	1	1	yes	characteristic
20	C₄×C₅×C₁₃	—	123	1	no	Hall
21	C₂×C₃×C₄₁	C₂×C₅×C₁₃	1	1	yes	characteristic
22	C₅×C₄₁	Dic₃×C₁₃	1	1	yes	characteristic, complemented, Hall
23	C₃×C₅×C₁₃	Dic₄₁	1	1	yes	characteristic, complemented, Hall
24	Dic₄₁	—	3	1	no	Hall
25	Dic₃×C₁₃	—	41	1	no	Hall
26	C₂×C₅×C₁₃	D₁₂₃	1	1	yes	characteristic
27	C₃×C₄₁	C₄×C₅×C₁₃	1	1	yes	characteristic, complemented, Hall
28	C₂×C₄₁	C₅×D₃×C₁₃	1	1	yes	characteristic
29	C₂×C₃×C₁₃	C₅×D₄₁	1	1	yes	characteristic
30	C₅×C₁₃	Dic₁₂₃	1	1	yes	characteristic, direct product factor, Hall
31	C₅×Dic₃	—	41	1	no	Hall
32	C₄×C₁₃	—	123	1	no	Hall
33	C₄₁	C₅×Dic₃×C₁₃	1	1	yes	characteristic, complemented, Sylow
34	C₃×C₁₃	C₅×Dic₄₁	1	1	yes	characteristic, complemented, Hall
35	C₂×C₃×C₅	C₁₃×D₄₁	1	1	yes	characteristic
36	C₂×C₁₃	C₅×D₁₂₃	1	1	yes	characteristic
37	C₄×C₅	—	123	1	no	Hall
38	C₃×C₅	C₁₃×Dic₄₁	1	1	yes	characteristic, complemented, Hall
39	C₁₃	C₅×Dic₁₂₃	1	1	yes	characteristic, direct product factor, Sylow
40	Dic₃	—	41	1	no	Hall
41	C₂×C₅	C₁₃×D₁₂₃	1	1	yes	characteristic
42	C₂×C₃	C₅×C₁₃×D₄₁	1	1	yes	characteristic
43	C₅	C₁₃×Dic₁₂₃	1	1	yes	characteristic, direct product factor, Sylow
44	C₄	—	123	1	no	Sylow
45	C₃	C₅×C₁₃×Dic₄₁	1	1	yes	characteristic, complemented, Sylow
46	C₂	C₅×C₁₃×D₁₂₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₅×C₁₃	D₁₂₃
commutator, hypocenter	C₃×C₄₁	C₄×C₅×C₁₃
Frattini	C₂	C₅×C₁₃×D₁₂₃
Fitting, socle, prime radical	C₂×C₃×C₅×C₁₃×C₄₁	C₂
cosocle	C₂×C₃×C₄₁	C₂×C₅×C₁₃
LCS	G ▷ C₃×C₄₁
UCS	C₁ ◁ C₂×C₅×C₁₃

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₃×C₅×C₁₃×C₄₁	C₄	semidirect
4. C₁₃×Dic₁₂₃	C₅	direct (N × H)
7. C₅×C₁₃×C₄₁	Dic₃	semidirect
8. C₅×Dic₁₂₃	C₁₃	direct (N × H)
10. C₃×C₁₃×C₄₁	C₄×C₅	semidirect
15. C₃×C₅×C₄₁	C₄×C₁₃	semidirect
16. C₁₃×C₄₁	C₅×Dic₃	semidirect
17. Dic₁₂₃	C₅×C₁₃	direct (N × H)
22. C₅×C₄₁	Dic₃×C₁₃	semidirect
23. C₃×C₅×C₁₃	Dic₄₁	semidirect
27. C₃×C₄₁	C₄×C₅×C₁₃	semidirect
30. C₅×C₁₃	Dic₁₂₃	direct (N × H)
33. C₄₁	C₅×Dic₃×C₁₃	semidirect
34. C₃×C₁₃	C₅×Dic₄₁	semidirect
38. C₃×C₅	C₁₃×Dic₄₁	semidirect
39. C₁₃	C₅×Dic₁₂₃	direct (N × H)
43. C₅	C₁₃×Dic₁₂₃	direct (N × H)
45. C₃	C₅×C₁₃×Dic₄₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃×C₅×C₁₃×C₄₁	2
C₅×C₁₃×Dic₄₁	3
C₁₃×Dic₁₂₃	5
C₅×Dic₁₂₃	13
C₅×Dic₃×C₁₃	41

Subgroups — C5×C13×Dic123 #44544285

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×C₁₃×Dic₁₂₃ #44544285